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I constantly see papers on "variational quantum algorithms" but I don't really see any explanation of what they are that are clear to me.

I found out about the variational method in quantum mechanics, which is an approximation to find the lowest energy eigenstate, but I don't see how this is related to quantum algorithms. (From this wikipedia page)

Does the variational method allow for stronger quantum computation? Do the algorithms solve problems outside of BQP?

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They are algorithms having a quantum part with parameters (for instance angles of rotations gates) which are optimized by a classical computer :

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Today's NISQ computers are noisy so algorithms using only full quantum processes can't give good results. Variational algorithms are a good alternative to take advantage of today's quantum computers.

Variational algorithm are used in Quantum Machine Learning for instance : the goal is to find the best parameters of what is called the ansatz circuit to get the best results (see for instance Variational Quantum Classifier).

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The review paper Variational Quantum Algorithms gives a comprehensive introduction and analysis on variational quantum algorithms.

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